Cuttingset methods for robust convex optimization with. Robust lp with righthandside uncertainty, duality and. Tilburg university adjustable robust optimization zhen, jianzhe. In contrast to ro, stochastic optimization starts by assuming the uncertainty has. Among the existing approaches where the uncertainty in. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. The course is covered by these lecture notes and more than covered by the book a. Model uncertainty, robust optimization and learning. Our contributions to the literature are as follows. The parameter varies between 0 and 1 and the optimization model is robust when all the uncertain coefficients can be realized within the bound provided by the uncertainty set 3. Recent works using this general approach include bental and nemirovski1998,1999,2000,bertsimasandsim2004.
Our focus will be on the computational attractiveness of ro approaches, as well. Originating from soyster 1973 and completely ignored for over two decades after its birth, the robust optimization was reborn circa 1997 and during the last decade became one of the most rapidly developing areas in optimization. Sep 21, 2011 one of the earliest papers on robust counterpart optimization is related to the work of soyster 4, who considered simple perturbations in the data and aimed at finding a reformulation of the original linear programming problem such that the resulting solution would be feasible under all possible perturbations. Robust optimization is a young and emerging eld of research having received a considerable increase of interest over the last decade. A robust optimization perspective on stochastic programming. On robust optimization journal of optimization theory and.
One is a sensitivitybased procedure, and the other makes. In general, the robust version of a tractable optimization problem may not itself be tractable. In this paper, we argue that the the algorithm engineering methodology ts very well to the eld of robust optimization and yields a rewarding new. One of the earliest papers on robust counterpart optimization is related to the work of soyster 4, who considered simple perturbations in the data and aimed at finding a reformulation of the original linear programming problem such that the resulting solution would be feasible under all possible perturbations. For negative results, which show that some robust optimization problems are for example nphard, see e. Robust optimization by melvyn sim submitted to the sloan school of management on may 14, 2004, in partial ful. Milton stewart school of industrial and systems engineering georgia institute of technology, atlanta georgia 303320205 usa november 2012.
Robust optimization with multiple ranges optimization online. Robust optimization refers to the modeling of optimization problems with data uncertainly. There are constraints with uncertain parameters that must be satis. Soyster 109, who was the first to consider, as early as in.
Robust optimization problems have first investigated by soyster 3 for linear. Columnwise uncertainty was first considered by soyster 9. Robust optimization is the approach that was chosen by bertsimas, and his robust optimization method from 2002 is based on it. We show how to derive both statements of a the dual to the robust model and b the robust version of the dual. The goal of the present chapter is to introduce a relatively new field, known as robust optimization, as an alternative to traditional methods and formulations.
Through an illustrative example, we suggest ways of putting robust optimization at work in environmental and energy optimization models. The basic premise is that by reformulating the original problem, or by solving a sequence of problems, we may find a solution which is robust to the uncertainty in the data. Enter your mobile number or email address below and well send you a link to download the free kindle app. This paper addresses the uncertainty problem in process scheduling using robust optimization.
Main contributors to robust optimization charnes, cooper et al. Worlds best practices for developing winning vehicles, is a ground breaking book whichintroduces the technical management strategy of robust optimization. There are several ways to apply robust optimization and the choice of form is typical of the problem that is being solved. Robust optimization with nonnegative decision variables. Secondly, two robust optimization models are established for the case when the demand volume of grain in each emergency site is uncertain. Nemirovski, robust optimization, princeton university press. The authors discuss what the strategy entails, 8 steps for robust optimization and robust assessment, and how to lead it in a.
For a recent account of the application of model uncertainty and robust optimization in economics and. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network. However, the realworld problems often deal with imprecise or ambiguous data. Soyster considers the linear optimization problem maximize c0x subject to xn j1. This is a direct application of not only the methodology of robust optimization, but also the motivation behind the development of robust optimization. Pdf data envelopment analysis under correlated uncertain. Cuttingset methods for robust convex optimization with pessimizing oracles almir mutapcic.
Jan 01, 2007 praise for robust portfolio optimization and management in the half century since harry markowitz introduced his elegant theory for selecting portfolios, investors and scholars have extended and refined its application to a wide range of realworld problems, culminating in the contents of this masterful book. Aug 10, 2007 praise for robust portfolio optimization and management in the half century since harry markowitz introduced his elegant theory for selecting portfolios, investors and scholars have extended and refined its application to a wide range of realworld problems, culminating in the contents of this masterful book. Then you can start reading kindle books on your smartphone, tablet, or. Uncertainty sets, formulations and probabilistic guarantees zukui li, christodoulos a.
Here one uses \uncertainbut bounded model of data perturbations, allowing the uncertain data to run through a given. A bound on the probability of constraint violation is derived. Robust convex optimization 1998 mathematics of operations research. This makes multirange robust optimization a novel theoretical extension of robust optimization with valuable practical applications. Some of the problem parameters are estimates and carry estimation risk. Robust optimization ro, on the other hand, is a class of techniques that does not need any information on the distribution of the uncertain parameters and was initiated by bental and nemirovski 1998. Robust discrete optimization and network flows springerlink. As a result, we believe that robust counterparts with nonnegative decision variables are needed.
Furthermore, as we will illustrate in 2, this problem is compounded by the fact that the deterministic forms of many robust optimization models are exceedingly complex and tedious to model explicitly. Robust optimization has developed as an alternative to stochastic programming to model decision making problems under conditions of uncertainty and to nd decisions that remain optimal over all scenarios by relying on worstcase scenario bounds. Robust optimization is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself andor its solution. Worlds best practices for developing winning vehicles is a musthave book for engineers and managers who are working on design, product, manufacturing, mechanical, electrical, process, quality area. One approach in these cases is to formulate a conservative approximation optimization. Applications of traditional data envelopments analysis dea models require knowledge of crisp input and output data. In its original form, proposed by soyster 71 and falk 48, robust optimization was mostly concerned with linear programming problems in which the data were inexact. Specifically, we show that a unique solution of a robust optimization problem is pareto optimal for the unconstrained optimization. Robust single objective optimization has been well studied. This approach dates back to soyster 1973, who considered a deterministic linear optimization model that is feasible for all data lying in a convex set. Because of the columnwise structure of the uncertainty considered, the robust optimization problem. Robust optimization is a minregret modelling methodology that seeks to minimise the negative impact of future events when the values of model parameters. Originating from soyster 1973 and completely ignored for over two decades after its.
Optimization for machine learning university of texas at. Robust optimization approximation for joint chance constrained optimization problem 21 may 2016 journal of global optimization, vol. The soyster s method converts the uncertainty problem into a regular linear programming problem by worst case scenario whereas the bentals and simons methods reduce the conservatism in the first method and find the optimal solution according to the. Robust models for location and inventory decisions in. Floudas department of chemical and biological engineering, princeton university princeton, nj 08544 abstract robust counterpart optimization techniques are studied in this paper. Distributionally robust optimization and its tractable. Compared to the traditionalscenariobased stochastic programming method, robust counterpart optimization method has a unique advantage, in that the scale of the corresponding optimization problem does not increase exponentially with the number of the uncertain parameters. We also refer the interested reader to the recent book of bental. It is shown that a particular robust counterpart of a nominal cilp can be reformulated as another cilp. An essential book for anyone working on optimization and decision making under uncertainty, robust optimization also makes an ideal graduate. Technical noteconvex programming with setinclusive.
On robust optimization journal of optimization theory. We also show that the robust counterpart of an nphard. Robust optimization ro is a relatively young methodology, developed mainly in the course of the last 15 years to analyze and optimize the performance of complex systems. Soyster 90 considered convex programming problems with setinclusive constraints, in which the worstcase approach is used to solve lps with box set uncertainty. Robust optimisation math bibliographies cite this for me. Abstract we consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. In this chapter, we propose a new approach for robust linear optimization that re tains the advantages of the linear framework of soyster 26. Using rome, we can readily model and solve a variety of robust optimization problems. The robust optimization method, which focused on treatability of computation in the case of data points disturbing in convex sets, was first proposed by soyster 2 and developed, respectively, by. Theory and applications of robust optimization 467 since there are more constraints to satisfy and the smaller the loss probability p loss. Robust portfolio optimization and management by frank j. The wide variety of applications and the fact that many problems in the class have been shown to be nphard, motivates the search for efficiently. Soyster s approach to robust optimization consider the linear program max ct x p6 subject to at i x.
In robust optimization, the parameter uncertainty is taken into account during the optimization. Brown y, constantine caramanis z july 6, 2007 abstract in this paper we survey the primary research, both theoretical and applied, in the. Our focus will be on the computational attractiveness of ro approaches, as. Pdf theory and applications of robust optimization researchgate. Model uncertainty, robust optimization and learning 2 informspittsburgh 2006, c 2006 informs 1994, garlappi, uppal and wang 2005, hansen and sargent 2001, 2001, 2003. Robust optimization and applications stanford university. Sims approach is the only one in the robust optimization literature which controls conservatism and applies to combinatorial problems with uncertainty present solely in the objective, as is. Robust optimization models can be useful in the following situations. On the other hand, the equilibrium problem provides a general mathematical model for a wide range of practical problems, such as optimization problems, nash equilibria problems, fixed point problems, variational inequality problems, and complementarity. A robust optimization framework for countably in nite linear programs cilps is developed. We show that in both cases the resulting robust optimization problem can be efficiently solved in. We introduce an unconstrained multicriteria optimization problem and discuss its relation to various wellknown scalar robust optimization problems with a finite uncertainty set. Robust optimization princeton series in applied mathematics 9780691143682. In section 2, we survey a wide array of optimization classes, and also uncertainty sets, and consider the properties of the robust versions.
Robust optimization is an emerging area in research that allows addressing different optimization problems and specifically industrial optimization problems where there is a degree of uncertainty in some of the variables involved. Robust multistage decision making stanford university. Theory and applications of robust optimization dimitris bertsimas. Robust optimization in countably in nite linear programs. That is, there is the need for robust formulations, which by virtue of rendering some parameters redundant in the classical robust optimization models yield solutions faster than the former. Robust optimization for process scheduling under uncertainty. Pachamanova third author is an expert, as well as some relatively advanced stat methods such as robust statistics and bayesian estimation. Robust portfolio optimization and management frank j. Soyster 1973 shows that the nonnegative sum of a finite. This paper covers the public release version of rome, version 1. Robust optimization made easy with rome 974 operations research 594, pp.
Both books discuss some latest optimization techniques however, this one has much more details on a modern method for handling uncertainty called robust optimization, in which dr. In this paper, we are going to introduce three methods to achieve robust optimization. Stochastic optimization starts by assuming the uncertainty has a probabilistic. Robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems. Aug 10, 2009 robust optimization is designed to meet some major challenges associated with uncertaintyaffected optimization problems. The present paper addresses the class of twostage robust optimization problems which can be formulated as mathematical programs with uncertainty on the righthand side coefficients rhs uncertainty. Brown y, constantine caramanis z may 31, 2007 abstract in this paper we survey the primary research, both theoretical and applied, in the. Lectures on robust convex optimization arkadi nemirovski. Robust linear programming with righthandside uncertainty. Robust optimization has become a powerful approach to handle uncertain optimization problems. This book is devoted to robust optimization a specific and relatively. Robust optimization can be viewed as a complementary to stochastic programming ap proach to handling optimization problems with uncertain data.
A tutorial on robust optimization, given at the ima. Convexprogrammingwithsetinclusiveconstraints and applications to inexact linear programming. The minicourse in question is aimed at overview of basic concepts and recent developments in this area. Robust optimization for environmental and energy planning. Twostage robust optimization, statespace representable. An essential book for anyone working on optimization and decision making under. Laguna, m, applying robust optimization to capacity expansion of one location in telecommunications with demand uncertainty. Central themes in ro include understanding how to structure the uncertainty set r with loss probability p loss.
Theory and applications of robust optimization citeseerx. Robust linear optimization under general norms sciencedirect. Robust optimization model for uncertain multiobjective. Aurelie thieley march 2006 abstract traditional models of decisionmaking under uncertainty assume perfect information, i. Specifically, we show that a unique solution of a robust optimization problem is pareto optimal for the unconstrained optimization problem. An essential book for anyone working on optimization and decision making under uncertainty, robust optimization also makes an ideal graduate textbook on the subject. Since model 3 is a nonlinear one, it is particularly not attractive for solving robust discrete optimization models. Practical robust optimization an introduction lnmb. Written by world renowned authors, robust optimization. Modern decisionmaking under uncertainty dimtris bertsimas. Robust optimization belongs to an important methodology for dealing with optimization problems with data uncertainty. Robust optimization is a relatively new and hot topic in the optimization community. One major motivation for studying robust optimization is that in many applications the data set is an appropriate notion of parameter uncertainty, e.1138 1058 903 1465 321 114 15 1585 273 537 1074 963 267 1038 333 1263 84 307 1212 1552 1271 1483 109 655 1134 391 140 143 870 676 246 30 233 538 404 195 914 533 1172 846 165 1140 357 1004 865